Problem:
Let be the least positive integer divisible by whose digits sum to . Find .
Solution:
Assume that there is such an less than , and let where , and are the digits of . According to the required properties, there is an integer such that and . Subtracting the second equation from the first gives . Thus is divisible by . If or , then or , respectively, and neither of these has digits that sum to . If , then , whose digits indeed sum to . Thus the requested integer is .
The problems on this page are the property of the MAA's American Mathematics Competitions